Introduction to Number System
The Number System is one of the most foundational chapters in Class 9 CBSE Mathematics. It builds the mathematical vocabulary every student needs from simple counting numbers to complex surds and exponents. This chapter forms the basis for algebra, geometry, and higher mathematics studied in Classes 10, 11, and 12.
This comprehensive guide covers all key concepts, definitions, formulas, and solved examples aligned with the NCERT/CBSE Class 9 syllabus.
Classification of Numbers
Understanding how numbers are classified is the first and most important step in this chapter.
1. Natural Numbers (N)
The set of all positive, non-fractional counting numbers starting from 1.
N = {1, 2, 3, 4, 5, …}
These are the most basic numbers used in everyday counting. They extend infinitely toward positive infinity.
2. Whole Numbers (W)
Natural numbers together with zero form the set of whole numbers.
W = {0, 1, 2, 3, 4, …}
Every natural number is a whole number, but 0 is a whole number that is not a natural number.
3. Integers (Z or I)
Integers include all whole numbers and their negatives — extending infinitely in both directions.
Z = {…, −3, −2, −1, 0, 1, 2, 3, …}
Integers include no fractional or decimal parts.
4. Rational Numbers (Q)
Any real number that can be expressed in the form p/q, where p and q are integers and q ≠ 0, is a rational number.
Examples: 2/3, 37/15, −17/19, 0.75, −2.333…
- All natural numbers, whole numbers, and integers are rational numbers.
- Rational numbers include terminating decimals (e.g., 0.75) and non-terminating but recurring decimals (e.g., 0.6̄, 0.45̄).
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Types of Fractions
| Fraction Type | Definition | Example |
|---|---|---|
| Common Fraction | Denominator is not 10 | 3/7 |
| Decimal Fraction | Denominator is 10 or a power of 10 | 3/10 |
| Proper Fraction | Numerator < Denominator | 3/5 |
| Improper Fraction | Numerator > Denominator | 5/3 |
| Mixed Fraction | Integral + Fractional part | 3 2/7 |
| Compound Fraction | Numerator and denominator are themselves fractions | (2/3)/(5/7) |
5. Irrational Numbers
All real numbers that cannot be expressed as p/q are irrational. Their decimal representations are non-terminating and non-recurring.
Examples: √2 = 1.41421356…, √3 = 1.73205080…, 2 + √3, ∛4
Note: √(−2) and √(−3) are not irrational they are imaginary numbers.
6. Real Numbers
Real numbers include all rational and irrational numbers. They can be represented on the number line.
The number line is a geometrically straight line with an arbitrarily defined origin (zero point).
7. Prime Numbers
Natural numbers with exactly two factors — 1 and the number itself.
P = {2, 3, 5, 7, 11, 13, 17, 19, 23, …}
How to identify a prime number:
- Find the approximate square root of the given number.
- Check divisibility by all prime numbers less than that square root.
- If none divide it evenly, the number is prime.
Example: Is 571 prime?
√571 ≈ 24. Primes below 24: 2, 3, 5, 7, 11, 13, 17, 19, 23.
571 is not divisible by any → 571 is prime.
8. Composite Numbers
Natural numbers that have factors other than 1 and themselves. All non-prime natural numbers (except 1) are composite.
C = {4, 6, 8, 9, 10, 12, …}
1 is neither prime nor composite.
9. Co-prime Numbers
Two numbers (not necessarily prime) are co-prime if their H.C.F. = 1.
Example: 4 and 9 → HCF(4, 9) = 1 → They are co-prime.
Any two consecutive integers are always co-prime.
10. Even and Odd Numbers
| Type | Definition | General Form | Example Set |
|---|---|---|---|
| Even | Divisible by 2 | 2n | {…, −4, −2, 0, 2, 4, …} |
| Odd | Not divisible by 2 | 2n − 1 | {…, −5, −3, −1, 1, 3, 5, …} |
11. Imaginary Numbers
Numbers whose square is negative.
Examples: 3i, 4i, i
where i = √(−1) (called iota)
Results:
- i² = −1
- i³ = −i
- i⁴ = 1
12. Complex Numbers
Numbers of the form Z = A + iB, where A is the real part and B is the imaginary part.
Examples: 2 + 3i, −2 + 4i, 11 − 4i
The set of complex numbers is the superset of all number sets.
Rational Numbers in Decimal Representation
Terminating Decimals
A finite number of digits appear after the decimal point.
Example: 1/2 = 0.5 ; 11/16 = 0.6875
Non-Terminating Recurring Decimals
A digit or group of digits repeats infinitely.
Examples:
- 2/3 = 0.6666… = 0.6̄
- 5/11 = 0.454545… = 0.4̄5̄
Converting Recurring Decimals to p/q Form
Example: Express 0.3̄ as p/q
Let x = 0.333…
10x = 3.333…
10x − x = 3
9x = 3 → x = 1/3
Finding Rational Numbers Between Two Numbers
Method 1 – Average Method
The number (a + b)/2 always lies between a and b. Repeat the process to find more rational numbers.
Method 2 – Equal Division Method
To find n rational numbers between a and b:
- Calculate d = (b − a) / (n + 1)
- The rational numbers are: a + d, a + 2d, a + 3d, …, a + nd
Example: Find 5 rational numbers between 3/5 and 4/5
d = (4/5 − 3/5) / (5 + 1) = (1/5) / 6 = 1/30
Answer: 19/30, 20/30, 21/30, 22/30, 23/30
Properties of Rational Numbers
| Property | Addition | Multiplication |
|---|---|---|
| Commutative | a + b = b + a | a × b = b × a |
| Associative | (a+b)+c = a+(b+c) | (a×b)×c = a×(b×c) |
| Identity | a + 0 = a (identity: 0) | a × 1 = a (identity: 1) |
| Inverse | a + (−a) = 0 | a × (1/a) = 1 |
| Distributive | a(b + c) = ab + ac | — |
Irrational Numbers
- The negative of an irrational number is irrational (e.g., −√3 is irrational).
- Sum/difference of a rational and irrational is always irrational.
- Sum/difference of two irrationals can be rational or irrational.
- Product of a non-zero rational and irrational is irrational (exception: 0 × √3 = 0, which is rational).
- Product of two irrationals is not always irrational.
Quick Examples:
| Operation | Result | Rational or Irrational? |
|---|---|---|
| √3 + (−√3) | 0 | Rational |
| (2+√3) + (2−√3) | 4 | Rational |
| √3 × √3 | 3 | Rational |
| 2√3 × 3√2 | 6√6 | Irrational |
| (2+√3)(2−√3) | 1 | Rational |
Surds
What is a Surd?
Any irrational number of the form ⁿ√a (read as "nth root of a") is a surd, where:
- a = radicand (must be a rational number)
- n = order of the surd
- ⁿ√ = radical sign
ⁿ√a can also be written as a^(1/n)
Expressions that ARE Surds
- ∛4 (radicand 4 is rational)
- 2√3 (surd + rational = surd)
- √(7 − 4√3) (since 7 − 4√3 = (2 − √3)² — a perfect square)
- ∛(√3) = ⁶√3 (can be simplified to a surd)
Expressions that are NOT Surds
- ∛8 = 2 (simplifies to a rational number)
- √(2 + √3) (radicand 2 + √3 is irrational)
- ∛(1 + √3) (radicand is irrational)
Laws of Surds
| Law | Expression | Example |
|---|---|---|
| Power Law | (ⁿ√a)ⁿ = a | (∛8)³ = 8 |
| Product Law | ⁿ√a × ⁿ√b = ⁿ√(ab) | ∛2 × ∛6 = ∛12 |
| Quotient Law | ⁿ√a ÷ ⁿ√b = ⁿ√(a/b) | ⁶√8 ÷ ⁶√2 = ⁶√4 |
| Nested Radical | ⁿ√(ᵐ√a) = ⁿᵐ√a | √(√2) = ⁴√2 |
| Change of Order | ⁿ√a = ⁿˣᵖ√(aᵖ) | ∛(6²) = ⁶√(6⁴) |
Surds can only be added or subtracted when they have the same order AND the same radicand.
Comparison of Surds
To compare surds of different orders, convert them to the same order using the LCM of their indices.
Example: Arrange √2, ∛3, and ⁴√5 in ascending order.
LCM of 2, 3, 4 = 12
- √2 = ¹²√(2⁶) = ¹²√64
- ∛3 = ¹²√(3⁴) = ¹²√81
- ⁴√5 = ¹²√(5³) = ¹²√125
Since 64 < 81 < 125: √2 < ∛3 < ⁴√5
Rationalization of Surds
Rationalization is the process of converting an irrational denominator into a rational one by multiplying with an appropriate Rationalizing Factor (R.F.).
Common Rationalizing Factors
| Surd | Rationalizing Factor | Product |
|---|---|---|
| √a | √a | a |
| ∛a | ∛(a²) | a |
| √a + √b | √a − √b | a − b |
| √a − √b | √a + √b | a − b |
| a + √b | a − √b | a² − b |
| ∛a + ∛b | ∛(a²) − ∛(ab) + ∛(b²) | a + b |
Conjugate surds (e.g., √a + √b and √a − √b) are rationalizing factors of each other. Sometimes the conjugate and reciprocal of a surd are identical (e.g., 2 + √3 and 2 − √3).
Geometrical Representation of Real Numbers on the Number Line
To represent √x on the number line:
- Mark point A on the number line.
- Mark point B such that AB = x units.
- From B, mark BC = 1 unit.
- Find midpoint O of AC.
- Draw a semicircle with centre O and radius OC.
- Erect a perpendicular at B meeting the semicircle at D.
- BD = √x
Use this method to plot √2, √3, √5, √6, etc. on the number line.
Exponents of Real Numbers
Positive Integral Exponent
For real number a and positive integer n:
aⁿ = a × a × a × … (n times)
Zero Exponent
a⁰ = 1 for any non-zero real number a.
Negative Integral Exponent
a⁻ⁿ = 1/aⁿ for any non-zero real number a.
Rational Exponent
For any positive real number a and rational number p/q:
a^(p/q) = (aᵖ)^(1/q)
Laws of Rational Exponents Formula
| Law | Formula | Example |
|---|---|---|
| Product Rule | aᵐ × aⁿ = aᵐ⁺ⁿ | 5² × 5⁴ = 5⁶ |
| Quotient Rule | aᵐ ÷ aⁿ = aᵐ⁻ⁿ | 5⁸ ÷ 5³ = 5⁵ |
| Power of a Power | (aᵐ)ⁿ = aᵐⁿ | (3²)³ = 3⁶ |
| Negative Exponent | a⁻ⁿ = 1/aⁿ | (3/4)⁻³ = (4/3)³ |
| Rational Exponent | a^(m/n) = ⁿ√(aᵐ) | 8^(2/3) = ∛(8²) = 4 |
| Product to Power | (ab)ᵐ = aᵐbᵐ | (2×3)² = 4 × 9 |
| Quotient to Power | (a/b)ᵐ = aᵐ/bᵐ | (2/3)³ = 8/27 |
Important Notes for Exam Preparation
- 1 is neither prime nor composite.
- 2 is the only even prime number.
- Any two consecutive integers are always co-prime.
- The sum/difference of two irrationals may or may not be irrational — always check.
- √(−2) ≠ −√2; √(−2) is an imaginary number equal to i√2.
- π and e are famous examples of irrational numbers.
- The set of complex numbers is the superset of all other number sets.
Formulas Table
| Formula Name | Mathematical Form | What It Means |
|---|---|---|
| Rational Number | p/q, q ≠ 0 | Expressible as a fraction of integers |
| Mean of Two Rationals | (a + b)/2 | Lies between a and b |
| n Rationals Between a & b | d = (b−a)/(n+1) | Step size for generating rationals |
| nth Root | ⁿ√a = a^(1/n) | Principal nth root of a |
| Rational Exponent | a^(p/q) = (aᵖ)^(1/q) | qth root of aᵖ |
| Product of Surds | ⁿ√a × ⁿ√b = ⁿ√(ab) | Only valid when order is same |
| Conjugate Surd | √a + √b ↔ √a − √b | Product gives rational number |
| Complex Number | Z = A + iB | A = real part, B = imaginary part |
| Iota Powers | i² = −1, i³ = −i, i⁴ = 1 | Cyclic pattern for powers of i |
| Difference of Surds | (√a + √b)(√a − √b) = a − b | Rationalization identity |